Wiener filter [9]

$$y[n]=x[n]+b[n]$$

$x[n]$ is the desired sequence and $b[n]$ is uncorrelated noise with both processes assumed to be wide-sense stationary. If one denotes

$$\hat{x}[n]=y[n]*h[n]$$

then the Wiener filter is the linear filter $h[n]$ that minimizes $(\hat{x}[n]-x[n])^2$. After some manipulation, this yields

$$r_{xy}[n]=h[n]*r_{yy}[n]$$

Since $x$ and $b$ are uncorrelated and WSS, one can write

$$r_{xy}[n]=r_{xx}[n] \quad r_{yy}[n]=r_{xx}[n]+r_{bb}[n]$$

This gives

$$r_{xx}[n]=h[n]*(r_{xx}[n]+r_{bb}[n])$$

which in the frequency domain is

$$H(\omega)=\frac{S_x(\omega)}{S_x(\omega)+S_b(\omega)}$$